About Global Existence for Quadratic Systems of Reaction-Diffusion

About Global Existence for Quadratic Systems of Reaction-Diffusion

Laurent Desvillettes, Klemens Fellner, Michel Pierre, Julien Vovelle November 10, 2006

Abstract: We prove global existence in time of weak solutions to a class of quadratic reaction-diffusion systems for which a Lyapounov structure ofLlogL- entropy type holds. The approach relies on an a priori dimension-independent L²-estimate, valid for a wide class of systems, and which provides anL¹-bound on the nonlinearities, at least for not too degenerate diffusions. In the more degenerate case, some global existence may be stated with the use of a weaker notion of renormalized solution with defect, arising in the theory of kinetic equa- tions.

Key Words: reaction-diffusion system, weak solutions, renormalized solu- tions, entropy methods

Mathematics Subject Classification: 35K57, 35D05

1 Introduction

To introduce the purpose of this paper, let us consider the following 4×4 reaction-diffusion system set on a regular bounded domain Ω ⊂ R^N: for i = 1,2,3,4,







∂_ta_i−d_i∆a_i= (−1)ⁱ[a₁a₃−a₂a₄] n·∇_xa_i = 0 on∂Ω

a_i(0) =a_i0≥0,

(1) where the di are positive constants, ai0 ∈ L^∞(Ω), and n denotes the outer normal to ∂Ω. The existence of a positive regular solution locally in time is classical. The global existence in time of a regular solution is not so obvious, and is even an open question in higher space dimensions. However, besides the preservation of positivity, this system offers some specificities which may be used to prove at least the existence ofglobal weak solutions in time.

Indeed, a main point is that the nonlinear reactive terms add up to zero.

Then, according to the Remark 2.2 in [PSch], it follows (by a duality argument) that the a_i are a priori bounded in L²(Q_T) for any T where we denote Q_T = (0, T)×Ω and for any dimension N. Consequently, the nonlinearities are a priori bounded in L¹(Q_T) for all T. Now, it follows from the results in [Pie]

that the above structure, together withL¹-bounds on the nonlinearities, provide the existence ofglobal weak solutions (see below for the meaning). We recall in the Appendix the main steps of this approach.

Here, we would like to extend the use of the dimension-independent L²- estimate just mentioned and show how it can be quite more exploited for this kind of systems and how it is robust enough to carry over to variable diffusion coefficients and even to degenerate diffusions coefficients. Let us explain our goals on the above specific system.

As it is well known, besides the property P

fi(a) = 0 (where we denote a= (a1, a2, a3, a4) andfi(·) is thei-th nonlinearity), it also satisfies the entropy inequality

X

i

log(a_i)f_i(a)≤0. (2)

As a consequence, if we denotezi=ailog(ai)−ai, one has

(z1+z2+z3+z4)t−∆(d1z1+d2z2+d3z3+d4z4)≤0.

Using the sameL²-estimate as the one just mentioned (see Appendix and The- orem 3.1), we can prove thatz =P

iz_i is bounded inL²(Q_T) for allT. This means that, not only the right-hand side of (1) is bounded in L¹, but it is uniformly integrable. Therefore, if we consider a good approximation of the sys- tem for which global existence in time of classical solutions holds, the nonlinear terms are uniformly integrable. On the other hand, by compactness properties of the heat operator, theL¹(QT)-bound of the right-hand side providesL¹(QT)- compactness of the approximate solution (aⁿ), and, up to a subsequence, con- vergence a.e. of the nonlinear terms. This, together with uniform integrability, yields convergence of the right-hand side in L¹. As a consequence, we obtain global existence for (1)using only theL²-estimates.

This is what we show below for a general class of systems for which a struc- ture of type (2) exists. Moreover, we show how the main L²-estimate may be extended to time-space dependent diffusions (and therefore to some quasi-linear problems) and even to some degenerate situations. It may also be applied to some classical quadratic Lotka-Volterra systems for which global existence of classical solutions is unresolved in high dimension (see [Leung],[FHM]).

In the last part of the paper, we provide some alternative when the nonlin- earities are not bounded inL¹. This is for instance the case when the diffusions are more degenerate. We then take up ideas aroundrenormalized solutionsfrom the theory of kinetic equations (see e.g. [DiL, CIP] for Boltzmann’s equation of gas dynamics). In particular, provided an L²-bound, we prove convergence of approximate solutions toward renormalized global solutions with defect measure (see e.g. [V2]).

2 Notations and general assumptions

All along the paper, we will use the following general notations and assump- tions: we denote byq ≥1 the number of equations of the system, and, for all i= 1, ..., q, we are given:

• di ∈ C¹([0,+∞)×Ω), di ≥ 0, with ∇x

√di ∈ L^∞(QT) (that is σ = 2P

ik∇x

√dik_L^∞_(QT)<∞for allT),

• f : (0,+∞)×Ω×[0,+∞)^q → R^q measurable and ”locally Lipschitz continuous”, that is: for f = (f₁, ..., f_q) and | · | denoting the Euclidean norm inR^q:

there existsk(·) : [0,+∞)→[0,+∞) nondecreasing such that a.e.(t, x)∈(0,+∞)×Ω,and∀r,rˆ∈[0,+∞)^q :

|f(t, x, r)−f(t, x,r)| ≤ˆ k(max{|r|,|ˆr|})|r−r|,ˆ and ∀T >0 : [(t, x)→f(t, x,0)]∈L^∞(QT),









 (3)

• and thepositivity preserving condition

a.e.(t, x),∀i,∀r∈[0,+∞)^q : f_i(t, x, r₁, ..., r_i−1,0, r_i+1, ...r_q)≥0.

For simplicity, we will often writef(r) =f(t, x, r), even whenf does depend on (t, x).

We will consider the following reaction-diffusion systems: for alli= 1, ..., q

∂tai− ∇x·(di∇xai) = fi(a),

n·∇xai = 0 [ or∀i= 1, ..., q, ai = 0 ] on ∂Ω.

ai(0) = ai0≥0.







(4) By regular solution on (0, T), we mean a function a ∈ C([0, T)×Ω) such that

∂_ta_i, ∂_xka_i, ∂_xkxla_i, f_i(a)∈L²(Q_T) and which satisfies the system pointwise.

Byweak solution, we mean a solution ”in the sense of the variation of con- stant formula”, that is,f(a)∈L¹(Q_T)^q for allT and

∀t≥0, a(t) =S(t)a0+ Z t

0

S(t−s)f(a(s))ds, (5)

whereS(t) is the linear semi-group associated with the linear part of the system with the same boundary conditions (that is,t→S(t)a₀is solution of the system withf ≡0 and the initial data a(0) =a₀).

For the definition of renormalized solutions, we introduce truncation func- tions T_k : [0,+∞) → [0,+∞) of class C², nondecreasing, concave and such that

∀r∈[0, k−1], T_k(r) =r, ∀r≥k, T_k(r) =k, ∀r,0≤T_k⁰(r)≤1. (6) By renormalized solution with defect measure, or ”renormalized superso- lution”, we mean a function a ∈ L¹(Q_T)^q with T_k⁰(a_i)f_i(a) ∈ L¹(Q_T) and T_k⁰(a_i)d_i∇xa_i∈L²(Q_T) such that, for allk >0 and all i= 1, ..., q

∂tTk(ai)− ∇x·(di∇xTk(ai))≥T_k⁰(ai)fi(a)−Tk00(ai)di|∇xai|². (7) If such a renormalized solution is regular enough so thatfi(a)∈L¹(QT) and di∇xai∈L¹(QT), then we may letk tend to +∞in (7) to obtain

∂tai− ∇x·(di∇xai)≥fi(a). (8)

Next, if on the other hand, the nonlinearity presents some kind of dissipative law like:

X

i

fi(a)≤0,

then we do obtain the reverse inequality in (8) for renormalized solution obtained as limits of regular solutions, and we are led to a (weak)-global solution (see Appendix and the proofs of Theorem 4.2 and Corollary 5.1 for such a two-sided approach).

Last remark: We will always considernonnegative solutions.

3 The main L²-estimate

The main result of this section is the following.

Theorem 3.1 Assume thatf satisfies the general assumptions of Section 2 and

∀r∈[0,+∞)^q, a.e.(t, x),

q

X

i=1

h⁰_i(ri)fi(r)≤Θ(t, x) +µX

i

ri,

whereΘ∈L²_loc [0,+∞), L²(Ω)

,µ∈[0,+∞)and fori= 1, ..., q

hi: [0,+∞)→[0,+∞) is convex continuous,∈C²(0,+∞), hi(0) = 0.

Let a be a regular positive solution of (4). Then, setting z_i = h_i(a_i), z = Pz_i, z_d=Pd_iz_i, we have

Z

Q_T

z zd≤C

kz(0)k²_L2(Ω)+

√

TkΘkL²(Q_T)

, (9)

whereC=C(µ, σ,maxi{kdik_∞}, T,Ω).

Remark: In the case of Dirichlet conditions, we may chooseC=C(Ω)e^2(σ2+µ)T (see (15),(16) in the proof). If moreover 0 =σ=µ= Θ, we obtain an estimate up toT = +∞, namely

min

i {di} Z

[0,+∞)×Ω

z²≤ Z

[0,+∞)×Ω

z zd≤C(Ω)kz(0)k²_L2(Ω).

This provides a first information for the asymptotic behavior ofa(t).

In all cases,zis bounded inL²(QT) in the nondegenerate case. It is interest- ing to notice that the productz zd is always bounded inL¹(QT) independently of a lower bound for thedi’s (that is to say, even if the system is degenerate).

We may slightly improve the dependence in z(0) in estimate (9) (see the Remark after the proof).

Proof: It is adapted from the particular case σ=µ= Θ = 0 (see [PSch] and also the Appendix which may be used in a first reading).

Usingh⁰⁰_i ≥0, we have for alli:

∂tzi− ∇x·(di∇xzi)≤h⁰_i(ai)fi(a),

so that

∂_tz− ∇x·(X

d_i∇xz_i)≤X

h⁰_i(a)f_i(a)≤Θ +µz. (10) Let us estimatezby duality. If we multiply the above inequation by somew≥0 regular enough, withw(T) = 0, and satisfying the same boundary conditions as thea⁰_is, we obtain

− Z

Ω

w(0)z₀− Z

Q_T

w_tz+w∇_x·(X

d_i∇_xz_i)≤ Z

Q_T

µzw+ Θw or also, after integration by parts

− Z

Ω

w(0)z0− Z

Q_T

wtz+X

zi∇x·(di∇xw)≤ Z

Q_T

µzw+ Θw, where we used

Z

∂Ω

wX

di∇xzi·n− ∇xw·nX

dizi = 0.

Thus Z

Q_T

Θw+ Z

Ω

w(0)z₀≥ − Z

Q_T

z(w_t+A∆w+B·∇xw+µw), (11) where we set

A:=zd/z, B:= (X

zi∇xdi)/z.

The dual problem: To estimate z by duality, we introduce the following dual problem whereH ∈ C₀^∞(Q_T) is an arbitrarynonnegative test-function and the boundary condition is the same as for thea⁰_is:

−(wt+A∆w+B·∇xw+µw) = H√ A, n·∇xw= 0,

resp. w = 0

on∂Ω, w(T) = 0.

(12) Thanks to the positivity of thez_i, we have

0≤min

i {d_i} ≤A≤max

i {d_i}.

If mini{di}>0, thanks to the regularity ofA, B, and up to changingtintoT−t, (12) is a good classical parabolic problem for which a unique positive solution w exists (see e.g. [LSU]). If A may vanish, we first replace it byA+, >0.

We will see that all the estimates that we are going to derive do not depend on minidi so that we may lettend to zero to solve the limit problem. Therefore, in what follows, we only make estimate on problem (12), assuming enough reg- ularity.

Withσ= 2P

ik∇x

√d_ik∞=P

ik∇xd_i/√

d_ik∞, we have

|B| ≤σ(X zi

pdi)/z=σ(X√ zi

√zi

pdi)/z≤σz^1/2z^1/2_d /z=σ√ A.

We deduce for the dual problem (12) that

−(wt+A∆w)≤√

A(σ|∇xw|+H) +µw. (13)

Multiplying (13) by−∆wand integrating over Ω give for allt∈(0, T):

−¹_2dt^d R

Ω|∇xw(t)|²+R

ΩA(∆w)²

≤R

Ω

√A|∆w|(σ|∇_xw|+H) +µR

Ω|∇_xw|². (14)

We set β(t) =R

Ω|∇xw(t)|². By Young’s inequality, the right-hand side of (14) may be bounded from above by

1 2

Z

Ω

A(∆w)²+ Z

Ω

σ²|∇xw|²+H²+µ|∇xw|². All this may be rewritten

−β⁰(t)−2(σ²+µ)β(t) + Z

Ω

A(∆w)²≤2 Z

Ω

H².

We deduce, settingρ(t) =e^2(σ2+µ)t, that

−d

dt(ρ(t)β(t)) +ρ(t) Z

Ω

A(∆w)²≤ρ(t) Z

Ω

2H². We integrate fromtto T and usew(T) = 0 to obtain

∀t∈(0, T), ρ(t)β(t) + Z T

t

ρ(τ)dτ Z

Ω

A(∆w)²≤ Z T

t

ρ(τ)dτ Z

Ω

2H²(τ), which implies

Z

Ω

|∇_xw(t)|²+ Z

[t,T]×Ω

A(∆w)²≤2ρ(T) Z

Q_T

H². (15)

Recall that, for someC=C(Ω) C

Z

Ω

|∇xw(t)|²≥ R

Ωw(t)² ifw= 0 on∂Ω

R

Ω(w(t)−_|Ω|¹ R

Ωw(t))² in all cases. (16) We can bound the averages ofw(t) by going back to the equation (12) and using that

R

Ωe^µtw(t) = −R

[t,T]×Ωe^µτ(wt+µw)

≤ e^µTR

Q_T|A∆w+B·∇xw+H√ A|

≤ p

maxikdik_∞C(T, µ, σ,|Ω|)(R

QTH²)^1/2,

(17)

so that we get in all cases sup

t∈[0,T]

Z

Ω

w(t)²≤C Z

QT

H², C=C(µ, σ,maxkdik_∞, T,Ω). (18) Back to the estimate ofz: We now come back to the inequality (11) which writes

Z

Q_T

zH√ A≤

Z

Q_T

Θw+ Z

Ω

w(0)z(0). (19)

Note thatR

Ωw(0)z(0)≤ kw(0)k_L2(Ω)kz(0)k_L2(Ω)and Z

QT

Θw≤ kΘk_L2(QT)kwk_L2(QT)≤ kΘk_L2(QT)

√ T sup

t∈[0,T]

kw(t)k_L2(Ω).

SinceH is arbitrary, we deduce by duality from (19),(18) that Z

Q_T

z zd=k√

Azk²_L2(QT)≤C[kz0k²_L2(Ω)+

√

TkΘkL²(Q_T)].

Remarks: Note that we actually get, not only anL²-estimate for the solution w of (12), but even ”maximal regularity” in L²(Q_T) for this equation in the sense that: ifH ∈L²(Q_T), then√

A∆w, w_tand∇xware separately inL²(Q_T).

We refer to the comments in [PSch] for the same questions inL^p.

We may improve the dependence on the initial data in Theorem 3.1 by using Sobolev imbedding in (16). Indeed, we may use instead

Z

Ω

w(0)^{p 2/p}

≤C Z

Ω

|∇_xw(0)|²+ Z

∂Ω

w(0)²

, (20)

forp= +∞ifN = 1, anyp <+∞ifN = 2 andp= 2N/(N−2) ifN ≥3.

As a consequence, using R

Ωw(0)z(0)≤ kw(0)kL^pkz(0)kL^q in the proof instead of an L²-duality, in Theorem 3.1, we may replace kz(0)kL²(Ω) by kz(0)k^2/q_Lq(Ω)

where

q= 1 ifN = 1, any q >1 if N = 2 andq= 2N/(N+ 2) ifN ≥3. This allows to solve the systems with weaker assumptions on the initial data.

4 Application to quadratic systems

Let us apply the above estimates to systems similar to the one given in the introduction, that is where thenonlinearity is at most quadraticand where the

”entropy” is controlled.

Theorem 4.1 Besides the assumptions of the introduction, assume that:

- the function k(·)in (3) satisfies k(r)≤C(|r|+ 1), - for all r ∈ (1,+∞)^q and a.e.(t, x), P

ilog(ri)fi(t, x, r) ≤ Θ(t, x) +µP

iri

whereΘ∈L²(QT), µ∈[0,+∞).

- ∃d0∈(0,+∞)such that,∀i= 1, ..., q, 0< d0≤di.

Then, the system (4) has a global weak solution in any dimension for all nonnegative initial dataa0 such that |a0|log(|a0|)∈L²(Ω).

Remark: As noticed at the end of the previous Section, we may relax the con- dition on the initial data to|a₀|log(|a₀|)∈L^q(Ω) for someq <2 well-chosen.

Proof of Theorem 4.1: We regularize the initial data and we truncate the nonlinearities fi by setting f_iⁿ(r) :=ψn(r)fi(r) where ψn(r) = ψ1(|r|/n) and ψ1: [0,+∞)→[0,1] isC^∞and satisfies

∀0≤s≤1, ψ1(s) = 1, ∀s≥2, ψ1(s) = 0.

Then, we easily check that the functionf_n satisfies also the assumptions of the introduction and of Theorem 3.1 (uniformly inn). Moreoverf_n is bounded on QT for all n. Therefore, by the classical theory of existence (see e.g. [LSU]), the approximate system has a unique regular global solution aⁿ on (0,∞) for regular approximationsaⁿ₀ of the initial dataa0.

We now apply Theorem 3.1 with h_i(x) = [xlog(x)−x]⁺. It follows that aⁿ_i log(aⁿ_i) is bounded inL²(Q_T) independently ofn. Since, the nonlinearityf is at most quadratic, it follows that fⁿ(aⁿ) is uniformly integrable onQT. By compactness of the linear operator in L¹(QT) (see e.g. [BP]), we may assume (up to a subsequence) that aⁿ converges as n → +∞ in L¹(QT) and a.e. to some a, this for all T. In particular, fⁿ(aⁿ) converges a.e. to f(a). But, uniform integrability onQT and convergence a.e. imply convergence inL¹(QT).

Consequently, we can pass to the limit in the formula aⁿ(t) =S(t)aⁿ₀+

Z t 0

S(t−s)fⁿ(aⁿ(s))ds, and this proves Theorem 4.1.

Remark: The above proof is rather simple thanks to the fact that the L²- estimate directly provides the uniform integrability of the nonlinearities. The situation is more delicate when one has only anL¹-bound on the nonlinearities.

Then, as explained in the Appendix, we may apply results from [Pie]. As a new example of the usefulness of theL²-estimate, we show here how one may prove global existence of weak solutions for quadratic Lotka-Volterra systems of the type described in [Leung] (see also [FHM]) and given as follows:

∂ta = D∆xa+AP(a−z)

∇xa·n = 0 on∂Ω, a(0,·) =a0(·)

(21) where the data are : D = diag{d1, ..., dq} a diagonal matrix with positive constants di, P = [pij] a q×q matrix and z ∈ (0,+∞)^q. The unknown is a: [0, T]→L²(Ω)^q andA= diag(a1, ..., aq). Herefi(a) =aiPq

j=1pij(aj−zj).

Theorem 4.2 Assume there existsΣ = diag(σ1, ..., σq)withσi>0 such that

∀w∈R^q, (Σw)^tP w=

q

X

i,j=1

σiwipijwj≤0. (22)

Then, the system (21) has a global weak solution on[0,+∞)for any nonnegative dataa₀∈L²(Ω)^q.

Proof: We obtain an a priori L²(Q_T)-estimate onaby applying Theorem 3.1 with

hi(r) =σi(r−zi)−σizilog(r/zi)f or r≥zi and hi(r) = 0f or r∈[0, zi].

Indeed, forri≥zi for alli, and by (22)

q

X

i=1

h⁰_i(ri)fi(r) =

q

X

i=1

σi(1−zi/ri)ri q

X

j=1

pij(rj−zj)≤0.

We use an approximation process as in the previous proof (f_iⁿ = ψ_nf_i which preserves the structure). Since the system is quadratic, the L²(Q_T)-estimate provides an L¹(Q_T)-bound on the nonlinear part of the approximate system.

According to the results in [Pie], up to a subsequence, the approximate solution

a_n converges a.e. and in L¹(Q_T) for all T to some function a which is a su- persolution of the problem. This means that there exist nonnegative measures µi, i= 1, ..., q onQT such that

∂tai−di∆xai=fi(a) +µi. (23)

Now, we use the fact that

∂t(X

i

σiaⁿ_i)−∆x(X

i

σidiaⁿ_i) =X

i

σif_iⁿ(aⁿ), (24) where, by (22)

X

i

σif_iⁿ(aⁿ)≤X

i,j

σizipij(aⁿ_j −zj).

We now pass to the limit in the sense of distributions in (24): we may use Fatou’s Lemma for the nonlinear terms, thanks to the previous bound from above which is linear with respect toaⁿ. We obtain

∂_t(X

i

σ_ia_i)−∆_x(X

i

σ_id_ia_i)≤X

i

σ_if_i(a).

But, together with (23), this proves that the measure P

iσ_iµ_i is equal to 0 and so is each µi so that the limit a is solution of the system in the sense of distributions.

To complete the proof, we also need to check that the initial data of a is indeed a0 and that the boundary conditions are preserved. For the Dirichlet conditions, we may use the bound on a in L¹(0, T;W₀^1,1(Ω)) coming from the L¹-bound on the right-hand side of the system (see e.g. [BP]). For the Neumann conditions, we repeat the above approach but with test functions in C^∞(QT) rather than only in C₀^∞(QT). Similarly, we control the initial data by using test-functions which do not vanish at t= 0. The details are left to the reader (see also [Pie]).

Remark: other choices of functions h_i. Theorem 3.1 may be used with other choices of convex functionsh_i:

1. hi(x) = σix with σi ∈ (0,+∞) is the simplest and corresponds to the fundamental case where P

ifi(r) ≤ 0. We then get an L²-estimate on the solution itself and global existence of weak solutions. Since we do not have in general uniform integrability of the nonlinearity, we act as in the previous proof.

2. hi(x) = x². This corresponds to the ”quadratic” Lyapunov structure P

irifi(r) ≤ 0. Theorem (3.1) says that the solution of (1) is then bounded in L⁴(QT) for all T. We may conclude to global existence as in Theorem 4.1 if the growth off(r) at infinity is strictly lower than|r|⁴. The limit case of growth|r|⁴ may be addressed as in the previous proof.

3. Similarly, the same will hold with ”h−subquadratic” systems satisfying P

ih⁰_i(r_i)f_i(r)≤ 0 and such that the growth of |f| at infinity is strictly less than|h|² (see the last Section).

5 Degenerate coefficients

Let us now consider the case of degenerate coefficients, for instance on the example given in the introduction, with variable C¹-coefficients, namely, for i= 1,2,3,4

∂tai− ∇x·(di∇xai) = (−1)ⁱ[a1a3−a2a4], (25)

∇_xa_i·n= 0 on∂Ω. (26)

We approximate the problem by regularizing the diffusions withdⁿ_i =di+n⁻¹. Existence of a solution aⁿ to the approximate problem is a consequence of Theorem 4.1. The main point is that, according to Theorem 3.1, we keep the uniform estimate

Z

Q_T

(X

i

z_iⁿ)(X

i

dⁿ_izⁿ_i)≤M (independent of n).

Theorem 5.1 Assume the di satisfy the assumptions of Section 2 and that

∃d0∈(0,+∞)such that

d₁+d₂+d₃+d₄≥d₀>0.

Assume a₀∈L²(Ω)⁴. Then,g_n(a) =aⁿ₁aⁿ₃−aⁿ₂aⁿ₄ is bounded inL¹(Q_T)for all T independently of n.

Remark: Obviously, the condition on the di’s allows that, for instance, three of them be identically equal to zero, the last one being bounded away from zero.

Or, they may all degenerate, as long as they do not all vanish at the same place.

Proof: We drop the indexation by n. We denote by M1, M2, ... positive constants independent ofn. SinceP

ifi ≤0, by Theorem 3.1, we have Z

QT

(X

i

ai)(X

i

diai)≤M1. This implies

Z

Q_T

d0min{a1a3, a2a4} ≤ Z

Q_T

(d1+d3)a1a3+ (d2+d4)a2a4≤M1. (27) Now, integrating the second relationPlog(ai)fi(a)≤0, we obtain

∀t∈(0, T), X

i

Z

Ω

|ailog(a_i)−a_i|(t) + Z

Q_T

|log a₁a₃/a₂a₄

||a1a₃−a₂a₄| ≤M₂.(28) This implies that, for the set K:= [a₁a₃≥2a₂a₄]∪[a₂a₄≥2a₁a₃],

Z

K

|a1a3−a2a4| ≤ Z

K

1 log 2

log(a1a3)−log(a2a4)

a1a3−a2a4

≤M2. The complement ofK isω₁∪ω₂ where

ω1= [a2a4≤a1a3<2a2a4], ω2= [a2a4/2< a1a3≤a2a4].